Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-8x+5y &= 5 \\ 7x-5y &= -1\end{align*}$
Answer: Begin by moving the $x$ -term in the second equation to the right side of the equation. $-5y = -7x-1$ Divide both sides by $-5$ to isolate $y$ $y = {\dfrac{7}{5}x + \dfrac{1}{5}}$ Substitute this expression for $y$ in the first equation. $-8x+5({\dfrac{7}{5}x + \dfrac{1}{5}}) = 5$ $-8x + 7x + 1 = 5$ Simplify by combining terms, then solve for $x$ $-1x + 1 = 5$ $-1x = 4$ $x = -4$ Substitute $-4$ for $x$ back into the top equation. $-8( -4)+5y = 5$ $32+5y = 5$ $5y = -27$ $y = -\dfrac{27}{5}$ The solution is $\enspace x = -4, \enspace y = -\dfrac{27}{5}$.